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Examination of Viscous Torque RelationshipsNathan D. Schiffrik

Physics Department, The College of Wooster, Wooster, Ohio 44691

April 29, 1998

Viscous torque, the resistance that a fluid offers to rotational motion, wasstudied with air on a polished metallic sphere. This experiment examineswhether the viscous torque, τ, is proportional to the angular velocity, ω, tothe first power, or to some higher power, by observing the rate of decay inangular velocity of a rotating sphere, subject only to viscous torque. Plotsof this decay rate on a semi-log scale produce linear results which showthat τ = −kω1. Additional data taken with flags to increase torquesuggests that for greater area, the viscous torque is actually proportional tothe square of the angular velocity. This experiment also examines therelationship between the viscous coefficient, k, and the area, A, exposed toviscous torque. Data appear to suggest a linear relationship between k andA, but results are insufficient to be conclusive.

INTRODUCTION

The resistance that a fluid offers tomovement is the fluid’s viscosity. Viscositymeasures “the internal friction that arises whenthere are velocity gradients within the system.”

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A common way to measure viscosity is to observeit when the viscous force is equal and opposite tothe gravitational force on a falling sphere, as in afalling sphere viscometer.

1 This method relates

viscosity to gravitation and demonstrates anexponential decay in velocity when the viscousforce is proportional to the velocity. Thisproportionality is related to Stoke’s law of theviscous force, f, on a sphere, which is f = 6πηvr,where η is the viscosity of the fluid, r is theradius of the sphere, and v is the constant velocitywith which it is falling.

1 However,

experimentation and observation can be madeeasier if the linear motion is translated intorotational motion. This can be done using agyroscope consisting of a sphere rotating in air.After being accelerated and then allowed to coast,the relationship between angular velocity andviscous torque can be easily observed.

If the viscous force is proportional to theangular velocity to the first power, then theangular velocity will decay exponentially in time.The viscous coefficient, k , is said to relateviscous torque to angular velocity in the followingmanner, τ = −kω . A new relationship can bestudied by adding areas of resistance, or rigidflags, over which the viscous force is acting, to

increase drag. The surface area added and theviscous coefficient can be observed and plottedagainst one another in an attempt to identify therelationship between them.

THEORY

The viscous force is proportional to the object’svelocity, v, giving the relationship F = −kv ,where k is the viscous coefficient. Relating thisequation to rotational motion results in

τ = −kω (1),where τ is the torque, ω is the angular velocity.Equating this equation to Newton’s 2nd law interms of rotational motion ( τ = Iα , where I is themoment of inertia, and α is the angularacceleration) and then solving the differentialequation for the angular velocity yields

ω = ωoe−λt (2).

where λ =k / I, which clearly shows how theangular velocity decays exponentially underviscous torque, at a rate proportional to theviscous coefficient k. Hence, -λ will be the slopeof a semi-log plot of angular velocity as itdecreases over time when subjected to a frictionalviscous torque.

Included in the viscous coefficient, k, aredependencies on the radius, r, of the sphere, andthe viscosity, η , of the fluid, as presented in thefollowing equation

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k = 6πηr (3).

2

This equation shows that the viscous coefficient,and hence λ as well, is proportional to theviscosity of the fluid, and is also linearly relatedto the radius of the sphere for linear motion.

However, the above theory is assuming alinear relationship between the viscous force andthe velocity. If, for the case of rotational motion,the viscous torque is proportional to any power ofthe velocity higher than the first power, thevelocity’s decay would not be exponential. As arepresentative example of all powers higher than1, the square of angular velocity is examined asproportional to the viscous torque, known as theNewtonian proportionality, as shown in thefollowing equation,

Idωdt

= −kω2 (4).

Following the same procedure of solving thisequation as was done to equation 11 yields

−1

ω+

1

ωo

= −k

It (5),

which when plotted as 1/ω versus t would give astraight line.

If the flags dominate in the second part ofthe experiment, the angular velocity is expected todecay at a rate greater than the linear Stoke’srelationship depicted in equation 2. With thegreater area exposed to the viscous torque, theangular velocity is expected to decay at a rateinversely proportional to time (t α 1 / ω), asdepicted by the Newtonian relationship inequation 5. Hence the linear relationship isexpected for a plot of 1 / ω versus t.

This experiment seeks to determine whichtheory holds for the relationship between viscoustorque and angular velocity for the sphere with norigid flags, that is to what power of the angularvelocity is the viscous torque proportional. Also,for cases with increased area due to added flags,this experiment seeks to determine if therelationship between viscous torque and angularvelocity changes due to the flags, and if so, whatis the new relationship. Also, the relationshipbetween the viscous torque and the area of theflags is investigated.

PROCEDURE

A 10 cm, polished, solid metal sphere with a longpole attached to it was used along with a basemount built to hold the rotating sphere. The basemount is suited with an input valve through whichcompressed gas can be pumped and releases underthe sphere to decrease the friction in rotation.

The angular velocity is determined byusing a He-Ne laser reflecting off the shinynorthern hemisphere where black stripes ofelectrical tape periodically interrupt the reflection,and from there reflect through a focusing convexlens onto a phototransistor. The Ne pressure thatfloats the sphere is set to a desired level of 10 psiso that the sphere is floating and is able to rotatewith only viscous friction. The filter on theHP5385A frequency counter is turned on toensure accuracy without external noisecontaminating the data, and the gate time on thefrequency counter is set at 1.0 second to helpensure the frequency counter is reading properly.

The LabView 3.1 program FrictionalTorque is launched and the gate timer on theprogram is set to record at 10 second intervals.After spinning the pole protruding from thesphere’s axis of rotation by hand up to amaximum frequency of approximately 7 Hz, readas 28 Hz on the frequency counter, and stabilizingthe pole to eliminate any wobble, the FrictionalTorque program was run to record data. Theprogram was allowed to run for approximately3000 seconds and stopped before any significantwobble could be observed that might skew thedata. The data was saved as a file and opened inIgor for analysis.

To investigate the relationship between theviscous coefficient and area, styrofoam flags ofincreasing area are attached to the axis pole of thesphere. The goal is to obtain a plot of the viscouscoefficient verses the area to observe how theviscous coefficient changes with respect toincreasing area. After attaching each flag, theprocess above was repeated and data was taken,always making sure to stop the run before anysignificant wobble appeared which could skewdata. Due to the inconvenience of workingaround the flags while spinning the pole, muchlower maximum frequencies were reached. Thesize of the flags used were (41.2±1.4)cm2,(81.8±1.5)cm2, and (121.4±2.3)cm2.

DATA and ANALYSIS

To determine the relationship between the viscoustorque and the angular velocity data runs weretaken with no flags put on,. The data was plottedand analyzed using Igor Pro. A sample of the firstdata run, as recorded by the computer program,can be seen in Table 1.

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Table 1. Sample of data from Run 1 (noflags), with time in units of seconds, frequencyin units of Hz, and the error for every timemeasurement being ±0.01%, and the error forevery frequency measurement being 1 partover (40*the frequency value).

time_s frequency47.94 6.0376558.42 6.0034668.94 5.963479.364 5.923989.833 5.88655100.273 5.84698

The graph of frequency versus time of the Run 1’sdata is shown in Figure 1.

Figure 1. Graph of Frequency versus Timefor Run 1 on a semi-log axis, with exponentialline fit. The exponential line fit fits the data sowell that it runs right through the middle ofmost of the data points and is hard to see.Over-laid on this graph is a graph ofFrequency versus Time for the run with twoflags, Flag 2, on a semi-log axis, withexponential line fit for initial decay down toan agular frequency ~1.7 s-1. The exponentialline fits the initial decay data well, yet the datashows some systematic deviation from the linefit at the later time and lower frequency.

When plotted on a semi-log axis, the data for therun without flags showed extreme linearity. Thismeans that the data is well explained by anexponential with constant decay rate. Anexponential line fit was performed to test thelinearity, and to find a value for the slope of theline which, as was shown in Figure 1, is theviscous coefficient, λ. Igor drew the fit lineaccording to the following equation

ω = k1 + k2e− k3t (6),which, if the variable k1 is forced to be zero onIgor, fits the form from equation 2, withωo

corresponding to k2 , and, as previously stated, k3

being the slope of the line corresponding to λ , theviscous coefficient. The values for run 1 were k2

= 6.214±0.001, k3 = (5.882±0.004)*10-4

.Data was taken with each size of flag

attached to the pole. The graph of the 2 Flagfrequency versus time can be found over-laid onthe graph of Run 1 in Figure 1. This portion ofthe experiment only investigates the relationshipbetween area and the viscous coefficient, so foreach data run with a different flag area, λ wasdetermined by finding k3 in the same manner as itwas found for the earlier runs. However, theexponential fit was performed, with k0 forced tozero, only on the initial decay, due to the data notbeing very linear when looked at as a whole. It isbelieved that perhaps the initial decay for thedifferent runs will provide a consistent approach.

The line fit for Flag 2 produced a value ofk3= λ = (4.0±0.1)*10

-3 s

-1, for the initial decay

down to an angular frequency ~1.70 s-1

. It isnoted that the data line is not at all straight, andalso that there is a small amount systematicdeviation of the data from the exponential line fitat later times and lower frequencies. The data ispossibly starting to show behavior consistent withtheories other than that of angular velocity to thefirst power.

As a further test of other theoreticalrelationships, the data for 2 Flag was taken and aplot was made of 1 / ω versus Time. A linear linefit was also then performed on the graph toobserve linearity predicted by equation 5.

1.0

0.8

0.6

0.4

0.2

1/ω

600500400300200100

Time (sec)

Figure 2. Graph of 1 / ω versus Time for therun with two flags , 2 Flag , with a linear linefit. The data is well explained by the linear fit,which implies the Newtonian form of theviscous torque is applicable.

The data produced a graph that illustrates verylinear behavior between 1 / ω and time. Thissupports the Newtonian theory depicted in

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7

8

91

2

3

4

5

6

Freq

uenc

y (r

evolu

tions

per

sec

)

300020001000Time (sec)

Run 1 (no flag)2 Flags

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equations 4 and 5, where the viscous torque isproportional to the square of the angular velocity.The linear fit produced a slope of(1.320±0.002)*10

-3 s

-2.

CONCLUSIONS AND DISCUSSION

One can observe, record, and infer the relationshipbetween frictional viscous torque on and angularvelocity of a smooth, rotating sphere from a plotof angular velocity, ω, versus time, t, and find it tobe consistent with the theory that the frictionalviscous torque is proportional to the angularvelocity to the first power. One can also observe,record, and plot the relationship between theviscous coefficient , λ , and the area of flagsattached and find the relationship to be linear assuspected from gathered data, although notenough data was gathered in this experiment tomake a conclusive statement on the relationship.

The frictional viscous torque was found tobe proportional to the angular velocity to the firstpower for the sphere rotating with no flags. Theviscous coefficient for angular velocity with noflags was found to be (6.0±0.2)*10

-4 s

-1. The

viscous coefficients for the 1 Flag of area(41.2±1.4) cm

2 , the 2 Flag of area (81.8±1.5)

cm2, and the 3 Flag of area (121.4±2.3) cm

2 were

found to be λ = (1.77±0.02)*10-3

s-1

, (4.0±0.1)10-3

s-1

, and (4.5±0.2)*10 -3

s-1

, respectively. Thegraphs of these data did not appear very linear ona semi-log plot for the data as a whole. Yet whenonly the initial decay was examined with a linearfit for each of the 3 flag runs, that region could bedescribed by a line corresponding to anexponential decay of the angular velocity, but thedata is much more consistent with a viscoustorque model that varies as the square of ω.

For further examination, the data for 2 Flag wastaken and a plot was made of 1 / ω versus Time totest the Newtonian theory that holds viscoustorque to be proportional to the square of theangular velocity. A linear line fit was also thenperformed on the graph, producing a slope of(1.320±0.002)*10

-3. The data fit the line fit

extremely well, indicating that with additionalarea added to increase drag, the viscous torque isin fact proportional to the square of the angularvelocity, and hence the connection to area is anopen question.

1 McGraw-Hill Encyclopedia of Science & Technology . 7thed. Vol. 19. (McGraw-Hill, New York, 1987). pp. 247-248.

2Reynolds, James and Stanton Hillier. “A Demonstration ofthe Proportional Relationship Between Frictional Torqueand Angular Velocity,” The Ealing Review. Nov./Dec.,1977. pp. 2-3.

Flewelling et al: Heat capacity anomaly in (CH3CH2)3N–H2O 5

J. Chem. Phys. Vol. 104, No. 20, 22 May 1996